An efficient branch-and-bound algorithm for submodular function maximization

The submodular function maximization is an attractive optimization model that appears in many real applications. Although a variety of greedy algorithms quickly find good feasible solutions for many instances while guaranteeing (1-1/e)-approximation ratio, we still encounter many real applications that ask optimal or better feasible solutions within reasonable computation time. In this paper, we present an efficient branch-and-bound algorithm for the non-decreasing submodular function maximization problem based on its binary integer programming (BIP) formulation with a huge number of constraints. Nemhauser and Wolsey developed an exact algorithm called the constraint generation algorithm that starts from a reduced BIP problem with a small subset of constraints taken from the constraints and repeats solving a reduced BIP problem while adding a new constraint at each iteration. However, their algorithm is still computationally expensive due to many reduced BIP problems to be solved. To overcome this, we propose an improved constraint generation algorithm to add a promising set of constraints at each iteration. We incorporate it into a branch-and-bound algorithm to attain good upper bounds while solving a smaller number of reduced BIP problems. According to computational results for well-known benchmark instances, our algorithm achieved better performance than the state-of-the-art exact algorithms.